On R&D forum there’s this active thread about that, but no good news at all (last comment was on 11 Aug by Polina, Veeam Team):

https://forums.veeam.com/veeam-backup-for-microsoft-365-f47/public-folder-backup-after-october-1st-2022-still-possible-t79413.html
 

I think a shared mailbox will be the “standard” workaround to this problem my friend.

Edit.

Ok, it’s the same link you posted 

Thx for your feedback @marcofabbri , but I already mentioned this forum-link in my post 

PS.
Microsoft surely wants to get rid of Exchange on-premise too after what happens in recent years… 

Yeah sorry, at first I didn’t click on link you posted.
My bad habit of reading too quickly...

No  problem @marcofabbri  

If possible, my recommendation is also the migration to shared mailboxes. This is what I have done at my old workplace. :)

Thx @Mildur .So, no other possibilities are possible to have a backup of public folders after the deprecation of basic authentication?

There is an expensive alternative. You could also install a DC and Exchange Server as Cloud VMs or On-Premise machines, do a Exchange Hybrid configuration and migrate the public folder from Exchange Online to the “Exchange Server”. Somehow. I’m not sure if there is a direct migration method. 

For me, the shared mailbox solution is currently the easiest and cheapest one. But I understand, it’s not easy for big enterprises with a lot of public folders and business processes around them.

Thx for your feedback @Mildur, that will not be an option 

I thought so

They’ve certainly been working on that...the next version of Exchange they noted a year or two ago, even on-premise, would be subscription-based which takes another half-step towards just migrating to the cloud.  

I’ve been working to move public folders to Shared mailboxes for users.  That said, I just cutover a M365 migration a week or so ago that has public folders as well, so that’s fun.  Fortunately, I think it’s just legacy data that’s not really needed very much, or not really needing to be backed up.  I’m not aware of a direct migration method from Public Folders to Shared Folders, but I will say that you could open up Outlook with access to both the old Public Folder and new Shared folder and more or less drag and drop, or more easily, export the public folder to a PST, and then import the PST to the shared folder.

Thx @dloseke for your feedback. Unfortunately we have a customer with almost 1TB of data in public folders (more than 20.000 !!!), they mis-used public folders as a kind of sharepoint . So we have to migrate the data from public folders to sharepoint online, not something that seems to be supported… Will be something nice

Yikes…..looks like CodeTwo has an app that may do it for you…..outside of a third-party app though, you’re in for a lot of manual labor I think.

https://www.codetwo.com/admins-blog/migrate-public-folders-to-shared-mailbox/

Ah….looks like MigrationWiz can do it too.  I wish I had realized that when I just migrated Public Folders to Public Folders…..could have migrated to a Shared Mailbox instead.  Although, I now just realized I remember seeing the prompt for that and didn’t think anything of it.  

https://help.bittitan.com/hc/en-us/articles/115008109707-Migrating-Public-Folders-to-Shared-Mailboxes

Thx, I know, but migrating to a shared mailbox or several will not be ideal because of size limit of 50GB and MS even recommended to use max. 50%. It will need to migrated to sharepoint online...

Oh...good point.  I hadn’t considered the sizing again.  I haven’t had anyone abuse public folders so badly in years…...that’s just very unfortunate.

That’s currently the way we suggest it to public folder lovers. Need not be to expensive: You get the remaining on-prem Exchange license key for free because you need a remaining one also if you just want to run remaining DCs in your on-prem environment.
Exchange hybrid licensing - Microsoft Q&A

If you host only a public folder database the, to my understanding there’s even no need for an additional paid licenses (as these would be user mailbox licenses anyways, which are not there). Correct me, if you have other intel.

Thought I would drop this Veeam KB here for reference. It has a link to a Microsoft article for migrating public folders to M365 Groups, rather than a shared mailbox. Perhaps something to consider.

The group mailbox size limit is still 50GB, although file storage seems to be inline with SharePoint limitations of 1TB. More info here.